#416702

order	512
nilpotency class	3
isoclinism	isoclinic class of #394329
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1248 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1248 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,5,7)(3,8,4,6)(9,10,12,13)(11,15,16,14)(17,18,20,22)(19,25,26,32)(21,29,27,24)(23,31,30,28);(1,3,5,4)(2,6,7,8)(10,13)(14,15)(18,22)(19,26)(21,27)(23,30);(1,4,5,3)(2,6,7,8)(10,13)(14,15)(17,19,27,31,20,26,21,28)(18,23,29,32,22,30,24,25);(9,11,12,16)(10,14,13,15)(18,22)(23,30)(24,29)(25,32)

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=ef,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323898);